📐 Estimation in Statistics

Last Updated: Jan 2026

Estimation is the process of using sample data to estimate an unknown population parameter.

Population parameters like:

• Mean ($\mu$)
• Proportion ($p$)
• Variance ($\sigma^2$)

are usually unknown, so we estimate them using samples.

🗣 Hinglish Tip: Estimation = sample ke basis par population ka value guess karna (scientific way me)

Types of Estimation

There are two main types:

1. Point Estimation
2. Interval Estimation (Confidence Interval)

Point Estimation

Point estimation gives a single numerical value as an estimate of a population parameter.

Common Point Estimators

Example: Point Estimation of Mean

• Sample data: 10, 12, 14, 16, 18
• Sample size: $n = 5$

Solution

$\bar{x} = \frac{10 + 12 + 14 + 16 + 18}{5}
\\[0.5em]
\bar{x} = \frac{70}{5}
\\[0.5em]
\bar{x} = 14$

$\checkmark$ Point estimate of population mean = 14

Limitations ⚠️:

• No idea about accuracy or reliability
• No range given

Interval Estimation (Confidence Interval)

Interval estimation gives a range of values within which the population parameter is likely to lie.

This range is called a Confidence Interval (CI).

Confidence Interval (CI)

A confidence interval is written as:

$Lower Limit < Parameter < Upper Limit$

Common confidence levels:

• 90%
• 95%
• 99%

🗣 Hinglish Tip: 95% CI = hume 95% confidence hai ki true mean is range ke andar hoga

Confidence Interval for Mean ($\sigma$ known)

Formula

$CI = \bar{x} \pm Z \left(\frac{\sigma}{\sqrt{n}}\right)$

Where:

• $\bar{x}$ = sample mean
• $\sigma$ = population standard deviation
• $n$ = sample size
• $Z$ = Z-value (from normal table)

Z-values (Common)

Example

• Sample mean: $\bar{x} = 50$
• Population standard deviation: $\sigma = 10$
• Sample size: $n = 100$
• Confidence level: 95%

Step 1: Identify Z-value

From table:

Z = 1.96

Step 2: Calculate Standard Error

$\frac{\sigma}{\sqrt{n}} = \frac{10}{\sqrt{100}} = \frac{10}{10} = 1$

Step 3: Calculate Margin of Error

$\text{Margin of Error} = Z \times \text{Standard Error}
\\[0.5em]
\text{Margin of Error} = 1.96 \times 1 = 1.96$

Step 4: Construct Confidence Interval

$\text{Lower Limit} = 50 - 1.96 = 48.04
\\[0.5em]
\text{Upper Limit} = 50 + 1.96 = 51.96$

$\checkmark$ Final Answer

$48.04 < \mu < 51.96$

$\checkmark$ We are 95% confident that population mean lies between 48.04 and 51.96

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